Dynamical Restoration of Z_N Symmetry in SU(N)+Higgs Theories

We study the Z_N symmetry in SU(N)+Higgs theories with the Higgs field in the fundamental representation. The distributions of the Polyakov loop show that the Z_N symmetry is explicitly broken in the Higgs phase. On the other hand, inside the Higgs symmetric phase the Polyakov loop distributions and other physical observables exhibit the Z_N symmetry. This effective restoration of the Z_N symmetry changes the nature of the confinement-deconfinenement transition. We argue that the Z_N symmetry will lead to time independent topological defect solutions in the Higgs symmetric deconfined phase which will play important role at high temperatures.

Z N symmetry Partition function of a pure SU(N) gauge theory at high temperature (T = 1 β ) is The allowed A's in the path integral are periodic in β, Contd...

S(A) and Z are invariant under the gauge transformation
V ( x, τ ) need not be periodic, as long as it satisfies the following eqn.
Z N is a symmetry of Z.

Order parameter of the theory
The Polyakov loop transforms nontrivially under Z N .
L is an order parameter for CD transition and it is analogous to the magnetization in a Z (N) spin system. Z N symmetry (with matter fields) The action in presence of fundamental Higgs field is given by, Being a bosonic field,Φ( x, 0) = Φ( x, β). Under above non-periodic gauge transformations,Φ (0) = Φ (β) (when z = 1 ).
It is not clear how this Z N explicit breaking will affect the CD transition. Fluctuations of the gauge and Higgs fields need to be considered.

Monte Carlo simulations of the CD transition
For simulations, we discretise the action on a 4D euclidean space, The discretised action is given by, Contd... In the Monte Carlo simulations an initial configuration of Φ n and U µ,n is repeatedly updated to generate a Monte Carlo history.
In an update a new configuration is generated from an old one according to the Boltzmann probability factor e −S taking care the principle of detailed balance.
Boltzmann factor and principle of detailed balance are implemented using pseudo heat-bath algorithm 1 2 for the Φ field and the standard heat-bath algorithm 3 for the link variables U µ 's.
To reduce auto-correlation between consecutive configurations we use over-relaxation method. In this Higgs phase diagram, the Higgs symmetric ( Φ = 0) and broken phase ( Φ = 0) are separated by the Higgs transition line. We compute the Polyakov loop distribution at various points on this phase diagram to study the Z N symmetry.

Results
Since the CD transition behaviour has been observed to be sensitive to N τ , we consider larger N τ for some values of the bare parameters.
Polyakov loop distribution (close to Higgs transition line) There is no Z 2 symmetry in the distribution H(L) of the Polyakov loop for SU(2).
Similarly for SU (3) there is no Z 3 symmetry of the Polyakov loop distribution.
Here Z N symmetry is explicitly broken.
Largest peak corresponds to the stable state and others correspond to meta-stable states. This is clear evidence that there is Z 2 symmetry.
This realization of the Z 2 symmetry makes the CD transition second order.
This symmetry restoration leads to critical behavior The value of the Binder cumulant(U L = 1 − L 4 3 L 2 2 ) at the crossing point for different volumes is consistent with the 3D-Ising Universality class.
It is clearly seen that, by scaling β g by t = ( The Z N symmetry is explicitly broken in the Higgs broken phase and close to the Higgs transition line in the Higgs symmetric phase.
Restoration of Z N symmety happens in the part of Higgs symmetric phase away from Higgs transition line. The Z N symmetry breaking line will approach Higgs transition line for Higher N τ . On the other hand, Higgs condensate decreases with decrease in κ.